Properties of similar triangles

Warm UpSolve each proportion.

1. 2.

3. 4.

AB = 16 QR = 10.5

x = 21 y = 8

Use properties of similar triangles to find segment lengths.Apply proportionality and triangle angle bisector theorems.

Objectives

Artists use mathematical techniques to make two-dimensional paintings appear three-dimensional. The invention of perspective was based on the observation that far away objects look smaller andcloser objects look larger.

Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings.

Example 1: Finding the Length of a Segment

Find US.

Substitute 14 for RU, 4 for VT, and 10 for RV.

Cross Products Prop.US(10) = 56

Divide both sides by 10.

It is given that , so by

the Triangle Proportionality Theorem.

Check It Out! Example 1

Find PN.

Substitute in the given values.

Cross Products Prop.2PN = 15

PN = 7.5 Divide both sides by 2.

Use the Triangle Proportionality Theorem.

Example 2: Verifying Segments are Parallel

Verify that .

Since , by the Converse of the

Triangle Proportionality Theorem.

Check It Out! Example 2

AC = 36 cm, and BC = 27 cm.

Verify that .

Since , by the Converse of the

Triangle Proportionality Theorem.

Example 3: Art Application

Suppose that an artist decided to make a larger sketch of the trees. In the figure, if AB = 4.5 in., BC = 2.6 in., CD = 4.1 in., and KL = 4.9 in., find LM and MN to the nearest tenth of an inch.

Example 3 Continued

Given

2-Trans. Proportionality Corollary

Substitute 4.9 for KL, 4.5 for AB, and 2.6 for BC.

Cross Products Prop.4.5(LM) = 4.9(2.6)

Divide both sides by 4.5.LM 2.8 in.

Example 3 Continued

2-Trans. Proportionality Corollary

Substitute 4.9 for KL, 4.5 for AB, and 4.1 for CD.

Cross Products Prop.4.5(MN) = 4.9(4.1)

Divide both sides by 4.5.MN 4.5 in.

Check It Out! Example 3

Use the diagram to find LM and MN to the nearest tenth.

Given

Check It Out! Example 3 Continued

2-Trans. Proportionality Corollary

Substitute 2.6 for KL, 2.4 for AB, and 1.4 for BC.

Cross Products Prop.2.4(LM) = 1.4(2.6)

Divide both sides by 2.4.LM 1.5 cm

Check It Out! Example 3 Continued

2-Trans. Proportionality Corollary

Substitute 2.6 for KL, 2.4 for AB, and 2.2 for CD.

Cross Products Prop.2.4(MN) = 2.2(2.6)

Divide both sides by 2.4.MN 2.4 cm

The previous theorems and corollary lead to the following conclusion.

Example 4: Using the Triangle Angle Bisector Theorem

Find PS and SR.

Substitute the given values.

Cross Products Property

Distributive Property

by the ∆ Bisector Theorem.

40(x – 2) = 32(x + 5)

40x – 80 = 32x + 160

Example 4 Continued

Simplify.

Divide both sides by 8.

Substitute 30 for x.

40x – 80 = 32x + 160

8x = 240

x = 30

PS = x – 2 SR = x + 5

= 30 – 2 = 28 = 30 + 5 = 35

Check It Out! Example 4

Find AC and DC.

Substitute in given values.

Cross Products Theorem

So DC = 9 and AC = 16.

Simplify.

by the ∆ Bisector Theorem.

4y = 4.5y – 9

–0.5y = –9

Divide both sides by –0.5. y = 18

Find the length of each segment.

1. 2.

Lesson Quiz: Part I

SR = 25, ST = 15

Lesson Quiz: Part II

3. Verify that BE and CD are parallel.

Since , by the

Converse of the ∆ Proportionality Thm.